Orbital mechanics of impulsive launch

ABSTRACT

Methods of launching a vehicle using impulsive force are disclosed. In one instance, a vehicle is launched easterly with impulsive force in a plane corresponding to the vehicle&#39;s elliptical orbital path. In another instance, a method of closing a timing difference is disclosed. The vehicle undergoes a series of forces after impulsive launch. The first force establishes an orbit having a period significantly different from the orbital period of a satellite or desired vehicle location, closing the difference in an integer number of orbits. The second force establishes the vehicle in circular orbit with the satellite or desired vehicle location. In another instance, the vehicle launched impulsively from a first celestial body travels a first path, and the vehicle experiences a second force along a hyperbolic path about the second celestial body and enters circular orbit about the second celestial body.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent applicationSer. No. 15/143,386, filed Apr. 29, 2016, and entitled “ORBITALMECHANICS OF IMPULSIVE LAUNCH,” inventor Harry E. Cartland, which isincorporated by reference herein as if put forth in full below.

BACKGROUND

Expansion of a light gas working fluid, e.g. hydrogen or helium, at hightemperature and pressure can accelerate projectiles to great velocitybecause of the fluid's very high sound speed, which is proportional (insimplest form) to the square root of temperature over molecular oratomic weight.

Prior work with light gas launch has largely focused on development ofhardware appropriate to this objective. For example, in U.S. Pat. No.8,979,033 (2015) and U.S. application Ser. No. 14/659,572 (2015), Hunteret al. disclose sea and land-based light gas launcher variants for spacelaunch applications. In U.S. application Ser. No. 14/642,720 (2015),Cartland describes the conditioning of the requisite large mass of lightgas to high temperature and pressure using a heat exchanger, i.e.without relying on adiabatic compression. In U.S. Pat. No. 8,536,502(2013) and U.S. Pat. No. 8,664,576 (2014), Hunter et al. disclosespinning and non-spinning space launch vehicles designed for hypersonicimpulsive launch, atmospheric egress, and orbital insertion.

In contrast to the prior hardware oriented work discussed above, theemphasis here is on methods, specifically the efficient orbitalmechanics or astrodynamics of orbital launch and rendezvous, as well asbeyond Low Earth Orbit (LEO) missions. Although developed for light gaslaunchers, the methods disclosed apply more generally to many impulsivespace launch concepts, including electromagnetic launchers, powder gunsand more exotic technologies.

A light gas launcher might at first appear somewhat constrained by itssize (and hence orientation) to a limited range of applications, butthis is not true. A mobile sea-based launcher can be moved to just aboutany launch latitude, and combined with flexibility in azimuth, canaccess the full range of orbital inclinations efficiently. In addition,both launch elevation and muzzle velocity can be adjusted to service anyorbital altitude efficiently. So a sea-based launcher is extremely agilein its ability to address a wide range of missions, no morekinematically constrained than a conventional rocket.

A land-based launcher is somewhat less agile, being limited by launchsite latitude and launcher orientation to fixed inclination (or a narrowrange of inclination), though still commanding some flexibility inaltitude. Even so, this isn't necessarily a shortcoming for certainapplications if the launcher is properly matched to the mission, as willbe discussed later.

SUMMARY OF THE INVENTION

The invention provides various methods of launching a vehicle using animpulsive force as applied by e.g. a light-gas gun, powder gun,electromagnetic launcher (e.g. coil gun), or other launcher thatprovides an impulsive force, examples of which are provided below.

In one instance, the invention provides a method of launching a vehicleto orbit around the Earth. An impulsive force launches the vehicle in adirection that is easterly or due east, and the vehicle ascends along atrajectory that defines an elliptical path having an apogee and aperigee. The launch direction for the vehicle is also in a plane thatcorresponds to the elliptical orbital path of the vehicle.

The invention also provides a method of closing a timing differencebetween a vehicle launched using an impulsive force and a satellite ofinterest or a desired vehicle location that was not yet attained. Aseries of forces is applied to the vehicle post-launch to provide achange in vehicle velocity Δv that is divided into a first Δv incrementand a second Δv increment. A first force of the series of forcesprovides the first Δv increment and temporarily places the vehicle intoa first orbit having a first orbital period that is significantlydifferent from an orbital period of the satellite or the desired vehiclelocation. This reduces a time difference between the vehicle and thesatellite or the desired vehicle location in an integer number oforbits. A second force of the series of forces provides the second Δvincrement, and the second force is sufficient to establish the vehiclein a circular orbit with the satellite or the desired vehicle location.

Further, the invention provides a method for a vehicle launch from afirst celestial body (such as the Earth) to a second celestial body(such as Mars). The method comprises using an impulsive force to launchthe vehicle and establish a first path for the vehicle. The method alsocomprises applying a second force to the vehicle along a hyperbolic paththat takes the vehicle from the first celestial body to the secondcelestial body, where the vehicle attains a circular orbit about thesecond celestial body.

Various advantages and details of these methods are discussed below.

TERMS OF REFERENCE

The analysis described here employs analytic spreadsheets, an analyticrendezvous model, and a full physics rocket code. The spreadsheets arebased on a ballistic missile formalism. The rendezvous model iterates asolution to the Kepler problem for the object trajectory, the Gaussproblem for the launch vehicle and object, and then the Kepler problemfor the launch vehicle fly out, employing a universal variableformalism. The spreadsheets and rendezvous model assume a rotatingspherical earth. Both incorporate an analytic drag loss based on anexponential atmosphere and constant drag coefficient, justified on thebasis that drag coefficients are flat at hypersonic speeds, the flightregime that obtains throughout. Both also assume instantaneous Δv,certainly valid for the initial impulsive launch and a reasonableapproximation for subsequent motor burns, which are relatively shortcompared to, mission time. The rocket code integrates the equations ofmotion and does not employ these approximations, though the morerigorous code results typically vary from the analytic approaches byless than 5% for LEO applications. These tools of analysis work intandem, the spreadsheets setting up the rendezvous model, which in turnprovides input to, and is refined by, the rocket code.

The methods disclosed apply to both land-based and sea-based launchers,although launch is assumed from sea level in the results shown here.Unless otherwise stated, muzzle velocity is 6 km s⁻¹, a good light gaslauncher working velocity consistent with minimal launch tube erosion.This launch speed also allows the employment of a single stage launchvehicle, important for reducing vehicle cost and complexity (andpotential for failure). Operation at a fixed muzzle velocity (or energy)may be advantageous from a mechanical perspective as stresses are alsofixed, although variation is easily possible; of course changing muzzlevelocity affects mission Δv requirement, arguably the most importantfigure of merit.

For representative launch vehicle properties, unless otherwise stated,the analysis assumes the characteristics of a single stage vehicledesigned in earlier work to place a 100 kg class payload in 53 deginclination, 770 km altitude circular orbit; they are launch mass m=1078kg (excluding sabot), drag coefficient C_(d)=0.0154 (power law shapewith base flare for passive stability), and ballistic coefficientβ=133,610 kg m⁻².

One of the most basic and useful missions is rendezvous off launchvehicle (vehicle) with an object of rendezvous (object) in LEO. Theobject may be:

1. a space station to be supplied

2. a depot to be stocked

3. a satellite to be serviced

4. an objective of military interest

5. something more exotic

where “something more exotic” could be, for example, an asteroidmaneuvered into orbit to be mined. The baseline object for this analysisis in a 51.65 deg inclination, 500 km altitude circular orbit; thiswould put it about 100 km above the International Space Station (ISS), areasonable depot location for support of beyond LEO manned spaceexploration. Of course the object may also orbit another body, e.g. theMoon or Mars.Basics

Assuming Δv is managed efficiently, in rough terms it takes a minimum ofabout 9 km s⁻¹ to reach LEO. Obviously it is possible to manage Δvinefficiently and expend more. Of this 9 km s⁻¹, about 7.5 km s⁻¹ is thecircular satellite velocity required to maintain. LEO, and about 1.5 kms⁻¹ is lost to drag and gravity.

Direction is important too, not just speed. Conic trajectories short ofescape originating at the Earth's surface later re-intersect the Earth.Even if the launcher could supply all the energy required to reach andsustain LEO, a means would still be necessary to change vehicledirection. So impulsively launched vehicles will in general have atleast one motor stage, although hypersonic aeromaneuvering using controlsurfaces is a possibility. With a 6 km s⁻¹ muzzle velocity, about 3 kms⁻¹ will have to come from the vehicle and elsewhere; under the rightcircumstances, Earth rotation can assist with several hundred meters persecond.

After some perhaps considerable reflection, the most efficient launchwill be easterly, directly into the desired orbital plane, withballistic apogee at the desired orbital altitude. The reasons for thisare addressed later. Efficiency is critical in that mass fraction, andhence useful payload, is reduced exponentially in proportion to theamount of Δv required from the vehicle.

The baseline orbit (500 km, 51.65 deg) can be reached with a 6 km s⁻¹launch to the east from 51.65 deg north latitude (inclination never lessthan launch latitude) at an elevation (initial flight path angle) ofabout 24 deg. Although irrelevant to the kinematics, a launch longitudeof 170 deg west places the launch location in the Aleutian Islands, apotential basing site for both land and sea-based launchers. Under theseconditions the vehicle must supply only about 2.63 km s⁻¹, with anadditional 0.29 km s⁻¹ deriving from Earth rotation.

What do these representative numbers mean? The conventional launchparadigm requires all of the Δv from the vehicle, which entailsexpensive, complex multistage rockets comprising mostly, engines andfuel, typically used once and then dumped in the ocean.

In contrast, impulsive launch at sufficient speed requires a relativelysmall vehicle Δv that can be obtained with a single stage, making for aless expensive, less complex, more reliable vehicle. Further, a small Δvallows a payload fraction of 2-3 tens of percent rather than the 2-3percent typical of conventional launch vehicles, even allowing for masspenalties associated with g-hardening and thermal protection. Most ofthe LEO velocity requirement comes from a reusable launcher (groundfacility) whose cost is amortized over many launches. All of theforegoing equates to a reduction in specific cost (dollars per kg) toLEO of an order of magnitude relative to conventional expendablemultistage launch vehicles. Recurring mission cost is a better metricfor small payload launch, and there is a similar comparative costreduction on that basis as well.

Impulsive launch becomes more cost efficient as the launch rateincreases since the capital investment in the facility is spread overmore missions. Hence there is an incentive to launch as frequently aspossible. Generally, there is only one “most payload efficient” launchopportunity as described above per day. Further, the timing becomesimportant too if the objective is rendezvous or placing a satellite at aprecise point in an orbit, an important consideration returned to later.With some loss in payload efficiency, two launch opportunities presentper day by launching from lower latitude, e.g. the equator, althoughthis requires reorientation of the launcher between launches, e.g. fromnortheast to southeast. Reorientation is facile enough with a sea-basedlauncher as there may be as many as 12 hrs to accomplish it; land-basedlaunchers are less flexible. The exception is launch from the equatorinto a zero inclination orbit, where a launch opportunity presents aboutever 90 min, or 16 times per day.

BRIEF DESCRIPTION OF FIGURES

Figures contained herein are not necessarily to scale and are providedto better illustrate aspects of the invention.

FIG. 1a shows the required Δv for insertion into circular Low EarthOrbits over a range of altitude for a 6 km s⁻¹ launch scenario.

FIG. 1b illustrates the concept of efficient indirect insertion.

FIG. 2a shows a most Δv efficient, direct ascent to rendezvous for thebaseline case with zero timing difference.

FIG. 2b shows a most Δv efficient, direct ascent to rendezvous for thebaseline case with a 5 min timing difference. The object leads thevehicle.

FIG. 3 shows a comparison of the vehicle ground track determined fromthe model and the rocket code for the baseline case.

FIG. 4a shows the required first burn Δv increment to correct a timingerror in integer N orbits for the baseline case when the object leadsthe vehicle.

FIG. 4b illustrates the concept of efficient timing correction.

FIG. 5 shows required vehicle Δv and launch time versus rendezvous timefor variation about the baseline case.

FIG. 6a shows the required total vehicle Δv versus launch time forrendezvous at the baseline rendezvous point.

FIG. 6b breaks the total Δv requirement shown in FIG. 6a into launch andrendezvous increments.

FIG. 7 shows required vehicle Δv and overall mission time as a functionof Earth departure velocity for direct ascent to 150 km Low Lunar Orbit.

FIG. 8 shows required vehicle Δv and overall mission time as a functionof hyperbolic excess velocity for a mission to 350 km Low Mars Orbit.

FURTHER DESCRIPTION OF PREFERRED EMBODIMENTS

Before proceeding with certain refinements, several digressions arehelpful. First consider Earth rotation. Although motion is not apparentto an observer in the Earth fixed frame, in the inertial frame Earthrotation adds an easterly component to the launch velocity equal to thetangential velocity of the Earth at the equator (464 m s⁻¹) times thecosine of the launch latitude. Hence Earth rotation provides the mostassistance for a launch azimuth due east—the easterly component of theEarth fixed launch velocity and the contribution from Earth rotation addas scalars—with increasing contribution as launch latitude decreasestoward the equator. Note also that for a launch due east, the orbitalplane accessed has an inclination equal to the launch latitude, thus thedesirability of an easterly launch from latitude matched to inclination.

Second, consider raising (or lowering) the altitude of an object in LEO.Although there are many ways to accomplish this, it is well known thatthe most Δv efficient means is by Hohmann transfer whereby a speedincrement in the direction of orbital motion puts the object on atransfer ellipse, which is then followed by a second increment about 45min (half an orbit) later at apogee of the ascent ellipse to matchcircular satellite speed at the new altitude. (Lowering an orbitproceeds similarly, where the increments are now braking burns, with thesecond occurring at descent ellipse perigee.) Although Hohmann transferis the slowest means of changing object altitude, it is very efficientin absolute terms, requiring only about 55 m s⁻¹ to effect an altitudechange of 100 km in LEO.

In fact, impulsive launch with circularization at ballistic apogee canbe understood in the context of Hohmann transfer. The launcher providesthe appropriate velocity on an ascent ellipse at the point where itcrosses the Earth's surface from a mathematical perigee tangential to acircular orbit well below the Earth's surface. For the baseline casediscussed here, that orbit has a radius of 1875 km, or an altitude of−4503 km. Since impulsive launch with circularization at ballisticapogee is effectively a Hohmann transfer, or at least part of one, therecan be no more efficient process for reaching a desired orbit altitude.

Third, consider plane change. Plane changes, unlike altitude changes,are expensive with respect to Δv required. For circular orbits, Δv isproportional to satellite velocity, and a plane change of only 1 deg forthe baseline case requires about 133 m s⁻¹. (Note that this is purely achange in inclination with no change in ascension.) In fact, a planechange of 60 deg requires a Δv equal to circular satellite velocity.Clearly it is highly advantageous to launch directly into the desiredorbital plane.

In summary, a launch to the east makes best use of Earth rotation,orbital insertion at apogee is most Δv efficient, and launch directlyinto the desired plane avoids expensive orbital maneuver. Sinceadditional Δv comes at the expense of dry mass fraction (payload), themost efficient launch will be easterly, into the desired orbital plane,with ballistic apogee matched to desired orbital altitude.

FIG. 1a shows the required Δv for insertion into circular orbits over arange of altitude for the baseline 6 km s⁻¹ launch scenario. The solidcurve shows a two pulse (burn) indirect insertion process, and thedashed curve a single pulse direct insertion process. The minimum in thesolid curve, and the intersection of the solid and dashed curves,correspond to the baseline requirement of 2.63 km s⁻¹ for a 500 kmcircular orbit.

In the indirect insertion the vehicle at apogee can provide a smaller Δvincrement than is necessary to circularize, putting it on a descentellipse to a perigee at lower altitude. A braking burn half an orbitlater establishes the vehicle, in an orbit at this lower altitude, andthe Δv required is nearly equal to the Δv saved by not circularizing atballistic apogee. In short there is almost no penalty for entering anorbit lower than ballistic apogee, as reflected in the flatness of thesolid curve in this region.

The vehicle can also be flown to a higher orbit than ballistic apogee,but there is no corresponding savings; Δv must be provided at ballisticapogee sufficient to circularize and enter an ascent ellipse, followedby a second burn in the direction of motion half an orbit later toestablish the higher orbit. In this case the vehicle pays the full costof raising the orbit; it must do the work the launcher did not.

Indirect insertion, illustrated in FIG. 1b , is another Hohmann-liketransfer and is very efficient. The direct insertion process is far lessso as the vehicle must negate some of its ballistic upward velocitycomponent, as well as provide sufficient tangential velocity to sustainthe orbit at that altitude. Direct insertion is, however, faster as theindirect process requires two burns approximately 45 min (half an orbit)apart.

FIG. 1a makes an important point. A large, fixed land-based launcher isnot so limited as it might first appear, at least with respect toaltitude. It can access a range of altitudes very efficiently, even moreso if muzzle velocity (i.e. ballistic apogee) is varied, Land-basedlaunchers are not as flexible as re-orientable sea-based systems, whichcan efficiently access a broad range of inclination as well, but thismay not be a drawback under certain scenarios such as stocking a depotwhere a dedicated launcher is matched to the orbit of the depot.

FIG. 2a shows an ideal, most Δv efficient, direct ascent to rendezvousfor the baseline case. The vehicle launches east from 51.65 deg northlatitude with ballistic apogee at 500 km altitude. The markers show 20 sintervals and time, zero is (arbitrarily) set to the point at which theobject ground track reaches maximum latitude. Note that since thevehicle initially moves more slowly than the object and must ascend toorbit altitude, it must launch well before the object passes (nearly)overhead. At the same time the launch site is rotating to the east, sofor the object to arrive over the same point in the inertial frame fromwhich the vehicle was launched, it will have to reach maximum latitudeat an Earth fixed longitude slightly to the west of the launch site. Therequired longitude difference is a function of the lead time of thevehicle and the Earth rotation rate. When optimal, the vehicle goes eastinto the same plane in the inertial frame as the object, which closesfrom behind, and the vehicle Δv increment is applied at apogee to matchobject speed. Practically speaking from a modeling perspective, therendezvous time, launch time, and launch longitude are iterated togenerate a requirement for a 6 km s⁻¹ muzzle velocity and east azimuth,where vehicle and object azimuth and flight path angle match atrendezvous, with a speed difference of 2.63 km s⁻¹. For the baselinecase, the model closes when the vehicle launches at −141 s withrendezvous at +311 s, and with the object ground track about 0.6 degwest of the launch site.

Note that the spreadsheet works “forward” from launch, while the modelworks “backward” from rendezvous. Both approaches are analytic and yieldidentical results. The dashed line in FIG. 2a is a more accurate vehiclefly out simulation using the rocket code. Although in good agreement, itshows that the analytic techniques over predict apogee altitude by about5%, almost certainly a result of approximations in computing drag loss.(The error relative to geocentric radius, the actual calculationalvariable, is much less.) FIG. 3 shows a comparison of the vehicle groundtrack determined from the model (markers) and the rocket code (dashedline); the agreement is excellent. Although the full physics code ismore rigorous, it is less useful than the model for surveying “parameterspace,” and comparisons indicate the model is accurate enough to bequite useful.

FIG. 2b represents a case when there is a timing difference between thevehicle and object. In this instance, by the time the launch siterotates into position to access directly the object orbit's plane, theobject is 5 min ahead of where it should be to effect a most efficientrendezvous. This is not unusual; a timing mismatch is the norm. Sinceideal conditions present infrequently, but a high launch rate isdesirable for economic and operational reasons, a means must bedetermined to mitigate timing error.

One method of correcting a timing error is initially to put the vehiclein an orbit whose period differs from that of the object. In the case ofFIG. 2b , the object leads the vehicle by 5 min, so the vehicle mustcomplete its orbit(s) faster than the object. Assume the object ofrendezvous is in circular LEO. With a first Δv increment (burn) atballistic apogee, insert the launch vehicle in a lower elliptical orbitof shorter period than the object, and then some integer number oforbits later initiate a second burn (at apogee) to put the vehicle incircular orbit in close proximity to the object with zero closingvelocity. Specifically, pick the first increment Δv₁ such that thetinting mismatch ΔT isΔT=N(T _(cs) −T)  Eqn. 1

where T_(cs) is the circular satellite period of the object, T is theperiod of the elliptical orbit of the vehicle, and N is an integernumber of orbits required to close the gap. It can be shown that therequired Δv₁ is

$\begin{matrix}{{\Delta\; v_{1}} = {{- v_{ba}} + {v_{cs}\lbrack {2 - ( {{NT}_{cs}/( {{NT}_{cs} - {\Delta\; T}} )} )^{\frac{2}{3}}} \rbrack}^{\frac{1}{2}}}} & {{Eqn}.\; 2}\end{matrix}$where v_(ba) is the speed at ballistic apogee and v_(cs) is the circularsatellite speed. The second velocity increment Δv₂ is simply the balanceof the Δv that would have been required to circularize at initialballistic apogee,Δv ₂ =v _(cs) −v _(ba) −Δv ₁  Eqn. 3so this method is no less efficient (to first order) than the idealtiming case of FIG. 2 a.

FIG. 4a shows the required Δv₁ for the baseline case to correct timingerror in N orbits for up to half an orbital period. The curves are cutoff at 2.544 km s⁻¹ since an initial Δv increment less than this amountwill put the vehicle on a descent ellipse with perigee under 200 km.Although this limit is somewhat arbitrary, drag on the vehicle in dieupper atmosphere becomes significant if it drops too low, and thevehicle will not return to ballistic apogee. FIG. 4a , or Equation 2,shows that the 5 min timing error of FIG. 2b can be corrected in twoorbits with Δv₁=2.561 km s⁻¹, or in five orbits with Δv₁=2.602 km s⁻¹,or in fifteen orbits with Δv₁=2.621 km s⁻¹. Closing the gap in fewerorbits puts Δv₁ on a steeper curve, making rendezvous more sensitive toerrors in burn time, although it will be faster. On the other hand,closing the gap in more orbits allows any errors or perturbations togrow for longer. So there will be some compromise, perhaps five orbitsin this instance.

Up to this point the focus has been on the instance where the objectleads the vehicle and the latter must catch up. It is also possible thatthe launch site rotates into position to access directly the objectorbit's plane too early, and the vehicle will lead the object. Using asimilar method, the vehicle can also fall back with a first Δv incrementgreater than that required to circularize, putting the vehicle in anascending elliptical orbit with a longer period than the object,followed after an integer number of orbits with a second braking Δvincrement at ballistic apogee (ellipse perigee), to circularize inproximity to the object. For “falling back” the vehicle pays a Δvpenalty, unlike the case of “catching up,” though this may be acceptablein time critical missions. One solution for the mission of stocking LEOdepots is to have multiple depots in the same orbit, so that there isalways one in fairly close proximity to which to catch up, rather thanexpending fuel to fall back. Note that the emphasis of this discussionhas been vehicle rendezvous with an object, but the same methods ofcourse also apply to, for example, simply positioning an impulsivelylaunched payload at some specific location in an orbit.

Note that in the derivation of Eqn. 2 above, ΔT was chosen as positivefor the case where the object leads the vehicle. In the instance wherethe vehicle leads the object, ΔT is negative and the second Δv incrementis a braking burn with magnitude the negative of that shown in Eqn. 3.

FIG. 4b illustrates the concept of efficient timing correction as may beapplied to various methods discussed herein.

The discussion of timing correction to this point has addressed the mostefficient scenario(s) wherein vehicle ballistic apogee matches theorbital altitude of the object or desired vehicle location. Lessefficient variations are possible, including those that vary muzzlevelocity as a launch parameter.

In the case where the launch opportunity comes too soon, e.g. the launchsite rotates into the plane of the intended orbit early, the muzzlevelocity can be increased to reach a ballistic apogee at an altitudeabove the intended orbit. At ballistic apogee, a first Δv incrementplaces the vehicle into a descending elliptical orbit with perigee atthe altitude of the object orbit or the desired location. The vehicleremains in this longer orbit for N+½ orbits (N=an integer) until theobject or desired location catches up, with the vehicle, at which pointa second Δv is applied at elliptical orbit perigee to establish thevehicle in orbit in close proximity to the object or location. Here, thesecond Δv is a braking burn.

In the case where the launch opportunity comes too late, e.g. the launchsite rotates into the plane of the intended orbit late, the muzzlevelocity can be decreased to reach a ballistic apogee at an altitudebelow the intended orbit. At ballistic apogee, a first Δv incrementplaces the vehicle into an ascending elliptical orbit with apogee at thealtitude of the object orbit or the desired location. The vehicleremains in this shorter orbit for N+½ orbits (N=an integer) until thevehicle catches up with the object or desired location, at which point asecond Δv is applied at elliptical orbit apogee to establish the vehiclein orbit in close proximity to the object or location. Here, the secondΔv is in the direction of vehicle motion.

These variations correct a timing error, but are less attractive thanthe baseline scenario(s) discussed previously, and illustrated in FIG.4b . In the first instance, the launcher provides excess kinetic energy,and a braking burn is eventually required to dispose of some of it. Inthe second instance the launcher does not provide the optimal kineticenergy, and the deficit is made up through expenditure of vehiclepropellant. Additional variations involving multiple burns and muzzlevelocity as a parameter are also conceivable.

Now consider some finer points and variations.

Because of the Earth's oblateness—it bulges about 21 km at theequator—an object's orbital plane precesses from gravitationally inducedtorque. This nodal regression westward (eastward for a retrograde orbit)amounts to about −5 deg per day for an object in a direct orbit at 52deg inclination and 500 km altitude, it means that the opportunity toaccess directly (and most efficiently) the object's plane comes about 20min sooner every day.

It was earlier mentioned that launching from latitude lower than themost Δv efficient baseline case would offer two opportunities per day toaccess, directly the object plane if the launcher can be reoriented.Consider the object ground track for the baseline case. It crosses theequator off the east coast of Sumatra (ascending node) and off the coastof Ecuador, between the mainland and the Galapagos. (Both of theselocations might be suitable for a sea-based launcher, although the downranges at the required azimuths may leave something to be desired.) Inthe former instance, an efficient rendezvous could have beenaccomplished with a 6 km s⁻¹ muzzle velocity at an Earth fixed azimuthof 34.2 deg with launch and rendezvous times of −1561 s and −1109 s,respectively. The latter instance would have required an azimuth of145.8 deg with launch and rendezvous times of +1278 s and +1729 s,respectively. Both instances would require only slightly more Δv atapogee (+13 m s⁻¹) than baseline, i.e. 2.64 km s⁻¹. The precise launchsite longitude is immaterial as long as timing correction is possible,and a single launcher at the equator will rotate into the object orbitalplane at both its ascending and descending nodes every day. Thus asingle launcher will have two launch opportunities a day, 12 hrs apart,to access the object plane with a reorientation through 111.6 degbetween (from northeast to southeast or vice versa)

Further Refinements

Not only is it necessary to determine parameters for most efficientlaunch, it is also important to understand how excursions from the idealaffect performance. How fast does the vehicle Δv requirement rise fromthe baseline case when rendezvous timing is not optimized? What does itmean to be “early” or “late”? There are (at least) two answers to thesequestions. The rendezvous may be early or late with respect to theoptimum rendezvous point in space and time. Or the vehicle may launchearly or late for rendezvous at the optimum point. (Or both may betrue.) Examine these separately.

First consider the effect on Δv for rendezvous at object position andtime other than optimum. FIG. 5 shows required vehicle Δv (solid curve)and launch time (dashed curve) versus rendezvous time. The minimum inthe solid curve at 2.63 km s⁻¹ corresponds to the baseline case withlaunch at −141 s and rendezvous at +311 s. Because muzzle velocity isfixed at 6 km s⁻¹, early rendezvous allows a later launch time as thevehicle has less distance to cover to the rendezvous point, and thevehicle arrives at rendezvous with positive flight path angle.Conversely, late rendezvous requires an earlier launch and more loftedfly out trajectory, and the vehicle arrives at rendezvous with negativeflight path angle. For very late rendezvous times (t>562 s),corresponding to very early launch times (t<−388 s), the flight pathangle at rendezvous is below the angle to the horizon and the vehicleapproaches the object against the background of Earth; in thiscircumstance “earthshine” could make rendezvous more challenging,especially if the object is not cooperative. The solid curve is fairlybroad near minimum and consequently deviation in rendezvous position oftens of seconds, perhaps 100 s, requires no significant increase invehicle Δv.

Now consider the second type of timing deviation, where the vehiclelaunch time varies from that necessary to yield the most efficientrendezvous, but with rendezvous at the optimum point. If the vehicle islaunched early or late with fixed muzzle velocity, it will have too muchor too little energy to arrive at the rendezvous point at the correcttime. So there are essentially two Δv increments required, one tocorrect the launch speed to ensure the vehicle arrives at the properpoint in space and time, and the second to match speed and directionwith the object at the rendezvous point.

FIG. 6b shows both vehicle Δv increments versus launch time, the launchΔv as a dashed curve and the rendezvous Δv as a solid one, (The launchΔv is assumed to be applied early, but outside the atmosphere where dragcan be neglected.) An early launch (t<−141 s) requires some braking soas not to arrive at the rendezvous point too soon, while a late launch(t>−141 s) requires augmentation of the muzzle velocity to arrive intime. The dashed curve reflects this, where Δv for early launch isnegative and, for late launch positive. (Of course the sign onlysignifies direction, and it is the absolute value that determinesvehicle Δv requirement.)

Now consider the rendezvous Δv (solid curve). For early launch, sincethe vehicle is braked so as not to arrive too soon, significant Δv(close to 4 km s⁻¹ for launch at −290 s) is required at rendezvous tomatch object speed. Conversely for late launch, since muzzle velocity isaugmented to arrive in time, the vehicle carries more speed to therendezvous point, and less Δv is then required to match object speed.For the most part, geometry at rendezvous is not much of a factor. Forthe range of launch time considered in FIG. 6b , vehicle and objectazimuth match closely at rendezvous, with some modest variation inflight path angle. The exception is at very late launch time, when thevehicle arrives with speed comparable to the object and flight pathangles deviate, requiring some Δv expenditure simply to correctdirection; this causes the upturn in the rendezvous Δv curve starting atabout −20 s.

FIG. 6a shows the total vehicle Δv requirement versus launch time, i.e.the sum of the absolute values of the components in FIG. 6b . Theminimum of this “checkmark” curve again corresponds to the baselinerequirement of 2.63 km for most efficient rendezvous. It is fairly broadas well and illustrates tolerance to deviation in launch tune of manytens of seconds.

Beyond LEO

The focus to now has been Earth orbit. In addition, impulsive launch canenable beyond LEO space exploration by, for example, staging materialsin Low Lunar Orbit (LLO) or Low Mars Orbit (LMO). By far the largestmission mass requirement for manned exploration of these destinations ispropellant, and pre-staging it reduces mission risk. Other essentialcommodities are compatible with impulsive launch as well, as are othertypes of payloads.

Consider first direct ascent from the Earth's surface to a direct LLO at150 km (positive specific angular momentum). Although absence of anatmosphere would seem to allow lower LLOs, below about 100 km they areunstable due to gravitational perturbations (with the exception ofcertain “frozen orbits” at specific inclinations). Assume a coplanartrajectory in the analytic patched conic approximation with the Laplacecriterion defining the transition point from the Earth's to the Moon'ssphere of influence. At a patch point, the vehicle state vector ismatched in two different reference frames, in this instance geocentricand selenocentric. The patched conic approximation provides goodestimates of mission requirements, although trajectory details are lessaccurate than for codes that integrate the three-body equations ofmotion because the vehicle moves for a period under significantinfluence of the Earth and the Moon simultaneously. An Earth departurevelocity at LEO altitude of less than about 10.6-10.9 km s⁻¹ leaves thevehicle with insufficient energy to reach lunar orbit and it falls backtoward Earth. Hence departure speed is close to Earth escape velocity(11.2 km s⁻¹) and a possible no return trajectory. Also, with the Moon'shigh tangential velocity (>1 km s⁻¹) and low escape velocity (2.4 kms⁻¹), lunar approach speeds are relatively fast. Consequently, departuretrajectories may be hyperbolic, while arrival trajectories are almostcertainly so. Any excess vehicle energy—that above minimumenergy—requires more Δv at departure and more braking on lunar arrival,making the trajectory more expensive on both ends, but the mission timeis shorter.

FIG. 7 shows required vehicle Δv (solid curves) and overall mission time(dashed curves) from launch to 150 km LLO insertion, as a function ofEarth departure velocity. The plots assume the baseline 6 km s⁻¹ muzzlevelocity with ballistic apogees at 300, 500 (baseline) and 700 km,corresponding to initial flight path angles (Earth fixed launchelevations) of about 18, 23 and 28 deg, respectively. Earth departureoccurs from near ballistic apogee at zero flight path angle, making mostefficient use of departure Δv. Launch azimuth is due east from 23 degnorth latitude, giving a representative inclination at the midpoint inthe range of the natural variation of the Moon's orbit relative to theequator (18.2 to 28.5 deg with 18.6 yrs period).

A minimum lunar mission Δv of 6.3-6.4 km s⁻¹ is required and is for themost part insensitive to departure altitude, a simple reflection ofconservation of energy. Starting lower in the Earth's gravity wellrequires a higher departure velocity, but more tangential velocity isavailable at ballistic apogee; these considerations offset. The minimumEarth departure Δv is 5.5-5.6 km s⁻¹, while LLO insertion requires about0.8 km s⁻¹ for braking. Additional Δv reduces mission time, verydramatically at first as the initial steepness of the dashed curvesillustrates.

Overall performance appears to slightly favor lower departure altitudeinsofar as it initially yields a shorter time-of-flight for theincremental amount of Δv expended. For example, a mission Δv of 7 kms⁻¹, about 0.7 km s⁻¹ above the minimum, leads to mission times of about44, 46 and 49 hrs for departures at 300, 500 and 700 km, respectively.

The most efficient means of travel from Earth to Mars is by Hohmanntransfer in the heliocentric frame, but that also has the longesttime-of-flight of any possible successful trajectory, and requires avery specific Earth departure phase angle such that Mars arrives at asweep angle of π rad at the same time as the vehicle. A more generaltrajectory, and one with a shorter time-of-flight is one that crossesthe orbit of Mars at some sweep angle less than π rad.

Many of the considerations applying to a Moon mission also apply to oneto Mars. In this case, the vehicle moves first primarily under theinfluence of the Earth, second under the influence of the sun, andfinally under the influence of Mars. So the patched conic approximationwill now have two patch points, first at the transition between thegeocentric and heliocentric frames, and second between the heliocentricand areocentric frames. The assumption is that the vehicle departs theEarth's sphere of influence in the direction of the Earth's motion inthe heliocentric frame (most efficient), and the departure velocityconsists mostly of the Earth's mean tangential velocity about the Sun(29.8 km s⁻¹). Mission Δv requirement and time-of-flight can becharacterized in terms of hyperbolic excess velocity, the residual speedin the geocentric frame after the vehicle has escaped Earth; in theorythis point is at infinity, though in practice is usually somesufficiently large distance from Earth, e.g. 1.5 million km. Earthdeparture trajectories are of course hyperbolic since with eccentricityless than one the vehicle would never escape the Earth. In contrast,heliocentric transfer trajectories are almost certainly elliptical as aneccentricity greater than one would imply sufficient energy to escapethe solar system, Mars arrival trajectories, like their lunarcounterparts, are hyperbolic. Finally, the Earth departure phase angleis an important parameter as the synodic period for Mars is 2.13 yrs,and a missed opportunity entails a long wait for the initial conditionsof a trajectory to repeat.

The Moon has a very slow rotation rate (period of 27.3 days), which whencoupled with its moderate radius (1738 km) means that there is no greatadvantage to a station or depot in a direct orbit versus one inretrograde from the perspective of rotational assist on ascent from thesurface. Mars is different. The rotation period of Mars is comparable tothat of Earth (1,026 days), and though its radius is more modest (3380km), the two coupled together lead to a tangential velocity at theMartian equator of 241 m s⁻¹. So the rotation of Mars can providesubstantial assistance to a vehicle leaving its surface, especiallyconsidering that Mars escape velocity is only 5.0 km s⁻¹. In fact,tangential velocity at the equator is a larger fraction of escapevelocity for Mars (4.8%) than for Earth (4.2%). Hence for Mars, astation or depot will most likely occupy a direct rather than retrogradeorbit, i.e. have positive specific angular momentum.

Another consideration for a Mars station or depot concerns altitude. ForEarth, the baseline LEO altitude was 500 km, putting it 100 km above 1SS. The Moon, lacking an atmosphere, allows a much lower altitude, e.g.LLO at 150 km. For Mars, although there are many factors to consider,LMO at 350 km is stable and puts a potential station or depot safelyabove the ionopause, mitigating potential problems for electronics. Thisis the representative altitude adopted here.

The assumption is that the vehicle starts its interplanetary trajectoryin the ecliptic plane. Since the Earth is inclined 23.4 deg to theecliptic, a most efficient launch is east from 23.4 deg north latitudeor 23.4 deg south latitude. The departure altitude, i.e. ballisticapogee, is 300 km, which requires an Earth fixed initial flight pathangle of about 18 deg for a baseline muzzle velocity of 6 km s⁻¹.

There are three significant components to the mission Δv. Most of therequired Δv is expended at injection and starts the vehicle on itstrajectory to Mars. And as in the lunar case, braking is necessary forinsertion into LMO. However, there is a third significant Δv requirementin this case: that for plane change from the ecliptic to that of Mars.For most of the planets in the solar system, these plane changes aresmall, but because the transfer velocities involved are large, the Δvcan be significant. The necessary 1.85 deg plane change is mostefficiently executed at a transfer trajectory true anomaly π/2 rad shortof arrival. Of course there are very likely other smaller Δvrequirements to, for example, make course corrections.

FIG. 8 shows required vehicle Δv (solid curve) and overall mission time(dashed curve) as a function of hyperbolic excess velocity for a missionto direct 350 km LMO. The total mission time is the sum of theballistic, Earth departure, interplanetary transfer and Mars arrivalflight times, which are determined from the appropriate form of theKepler equation for each phase of flight. The departure phase angles forthe range of hyperbolic excess velocity shown here are about 40 deg.With a muzzle velocity of 6 km s⁻¹, it takes a minimum of about 9 km s⁻¹from the vehicle to reach LMO, broken down into about 6 km s⁻¹ forinjection, 1 km s⁻¹ for plane change, and 2 km s⁻¹ for insertion. Asexcess hyperbolic velocity increases, the time-of-flight initially dropsrapidly, but reaches a point of diminishing returns. At the same time,the total Δv requirement rises steadily since injection Δv increaseswith hyperbolic, excess speed, plane change Δv increases with transfervelocity, and insertion (braking) Δv increases with arrival velocity.

With 6 km s⁻¹ muzzle velocity, the launcher provides about two-thirds ofthe total velocity requirement for LEO, almost half of, the minimum forLLO, and around 40% of the minimum for LMO. This Δv savings is extremelysignificant. Because of the inverse exponential dependence of massfraction on Δv, it translates (very approximately) to a mass fractionenhancement of a factor of five to ten depending upon assumed propulsionsystem efficiency, a factor of about five for liquid oxygen/liquidhydrogen (I_(sp)≈390 s), or a factor of ten for ammoniumperchlorate/aluminum (I_(sp)≈270 s).

The launch destination may be a LaGrange point of two celestial bodies,where one of these celestial bodies is in a circular orbit about theother celestial body. LaGrange Points are metastable or stable points inspace resulting from three-body effects that have been proposed as depotlocations to support beyond. LEO space exploration. There are fiveLaGrange points L1 . . . L5 that are either stable or metastable. Forexample, L1 is a metastable point between Earth and Sun, located 1.5million km from Earth, where the gravitational influence of Earthcounteracts that of the Sun to the extent that an object located at L1can maintain its position relative to both bodies. L4 and L5 are stablepoints about which an object can orbit, and in the Earth-Sun system arelocated in the plane of motion of the Earth about the Sun at theEarth-Sun separation distance from both (equilateral triangle), oneleading and the other following the motion of the Earth about the Sun,L4 and L5 points, being stable, are known to trap objects, while theother LaGrange Points are metastable and require stationkeeping. SimilarLaGrange points also exist for the Earth and the Moon, for instance,LaGrange points can be accessed through impulsive launch as well, in thesame manner as the methods described above to reach celestial bodiesfrom Earth.

Variations

Many variations on the foregoing are possible and will be obvious tothose of ordinary skill.

A satellite may, of course, be any sort of natural or man-made spaceobject. A natural satellite may be a body that has either orbitedanother body for a substantial period of time (such as the Moon orbitingEarth), or a natural satellite may be a body that has been captured andmoved to orbit (e.g. a comet or asteroid that has been moved into Earthorbit). A man-made satellite may of course be any of a large number ofman-made objects (e.g. a vehicle such as a crewed capsule; acommunications satellite; an observational satellite; a supply vehiclecarrying supplies such as fuel, equipment, construction supplies, parts,or food and beverages; or a space station).

The methods disclosed herein may be used to place a vehicle on a path ortrajectory that includes any sort of orbit. Any of the orbits for thevehicle, satellite, and place of rendezvous may be circular or may beelliptical, for instance. Such orbits include, without limitation, lowearth orbit, geosynchronous orbit, geostationary orbit, andsun-synchronous orbit. The path: or trajectory, especially the launchtrajectory, may or may not place the vehicle on an orbital path thatintersects with the surface of the body about which the vehicle is toorbit. A second force will of course be applied to the vehicle to changeits orbital path to prevent the vehicle from intersecting with thebody's surface in the methods described herein.

The forces applied to a vehicle according to any of the methodsdiscussed herein are typically not exclusively the forces that areapplied to the vehicle. Other forces such as correctional forces toestablish or maintain a desired orbit, stable orbit, desired or stableorientation, or desired trajectory may be applied.

Consequently, what is disclosed herein includes, without limitation onthe scope of the invention described above, the following:

-   -   1. A method of launching a vehicle to orbit about Earth, wherein        the method comprises launching the vehicle in a direction that        is easterly using an impulsive force and along a trajectory that        defines an elliptical orbital path,        -   a. wherein the elliptical orbital path has an apogee and a            perigee and        -   b. wherein the direction of launch is in a plane            corresponding to the elliptical orbital path of the vehicle.    -   2. A method according to paragraph 1 wherein the direction is        due east.    -   3. A method according to paragraph 1 or paragraph 2 wherein the        vehicle's trajectory and an orbit of a space object are in the        same plane.    -   4. A method according to any one of paragraphs 1-3 wherein        -   a. the vehicle's trajectory apogee is closely matched to (1)            an orbital altitude of a space object or (2) a place of            rendezvous with the space object,        -   b. wherein the vehicle has a fly-out time from a launch            site, said fly-out time being measured from a launch time to            a time that the vehicle first achieves the vehicle's            trajectory apogee, and        -   c. wherein the vehicle launch occurs about one-third of said            fly-out time prior to the space object passing overhead of a            position of the launch site at the launch time.    -   5. A method according to any one of paragraphs 1-3 wherein the        method further comprises applying a first force to the vehicle        at the vehicle's trajectory apogee, said force being less than a        force needed to establish a circular orbit for the vehicle so        that the vehicle enters a second and descending elliptical        trajectory which has a perigee at an altitude above the Earth        that is lower than an altitude of the vehicle trajectory apogee.    -   6. A method according to paragraph 5 wherein the method further        comprises applying a second force to the vehicle at the second        elliptical trajectory perigee to establish a circular orbit        having an altitude lower than the altitude of the vehicle's        trajectory apogee.    -   7. A method according to any one of paragraphs 1-3 wherein the        method further comprises applying a first force to the vehicle        at the vehicle's trajectory apogee so that the vehicle enters a        second and ascending elliptical trajectory having an apogee at        an altitude above the Earth that is higher than an altitude of        the vehicle's trajectory apogee.    -   8. A method according to paragraph 7 wherein the method further        comprises applying a second force to the vehicle at the second        elliptical trajectory apogee to establish a circular orbit at an        altitude above the Earth that is greater than the altitude of        the vehicle's trajectory apogee.    -   9. A method according to any one of paragraphs 1-3 wherein the        method further comprises applying a first force to the vehicle        at the vehicle's trajectory apogee to establish a circular orbit        for the vehicle.    -   10. A method of closing a timing difference between a vehicle        launched using an impulsive force and a satellite of rendezvous        or a desired vehicle location, comprising applying a series of        forces to the vehicle to provide a change in vehicle velocity Δv        that is divided into a first Δv increment and a second Δv        increment, wherein        -   a. a first force of the series provides the first Δv            increment and temporarily places the vehicle into a first            orbit having a first orbital period that is significantly            different from an orbital period of the satellite or the            desired vehicle location so as to reduce a time difference            between the vehicle and the satellite or the desired vehicle            location in an integer number of orbits, and        -   b. a second force of the series provides the second Δv            increment and is sufficient to establish the vehicle in a            circular orbit with the satellite or the desired vehicle            location.    -   11. A method according to paragraph 10 wherein        -   a. the vehicle follows a path of an elliptical trajectory            having a ballistic apogee and a ballistic perigee;        -   b. the first force is applied to the vehicle at the            ballistic apogee to establish a second elliptical orbit            having a second apogee and a second perigee, the second            elliptical orbit being a descending elliptical orbit,            wherein the second apogee has an elevation equal to an            elevation of the ballistic apogee, and        -   c. the second force is applied when the vehicle is at the            second apogee.    -   12. A method according to paragraph 10 wherein        -   a. the vehicle follows a path of an elliptical trajectory            having a ballistic apogee and a ballistic perigee;        -   b. the first force is applied to the vehicle at the            ballistic apogee to establish a second elliptical orbit            having a second apogee and a second perigee, the second            elliptical orbit being an ascending elliptical orbit,            wherein the second perigee has an elevation equal to an            elevation of the ballistic apogee, and        -   c. the second force is applied when the vehicle is at the            second perigee,    -   13. A method according to paragraph 11 or paragraph 12 wherein        the second force additionally matches the vehicle velocity to a        velocity of the satellite or the desired vehicle location.    -   14. A method according, to any one of paragraphs 11-13 wherein        the first Δv increment is selected to satisfy the equation        ΔT=N(T_(cs)−T) where ΔT represents a timing difference between        the satellite or the desired vehicle location and the vehicle,        T_(cs) is a period of the satellite's orbit, T is a period of        the elliptical orbit of the vehicle, and N is an integer number        of orbits required to correct a distance between the vehicle and        the satellite or the desired vehicle location.    -   15. A method according to paragraph 14 wherein the first Δv        increment has a value

${\Delta\; v_{1}} = {{- v_{ba}} + {v_{cs}\lbrack {2 - ( {{NT}_{cs}/( {{NT}_{cs} - {\Delta\; T}} )} )^{\frac{2}{3}}} \rbrack}^{\frac{1}{2}}}$where v_(ba) is vehicle speed at ballistic apogee and v_(cs) is a speedof the satellite in circular orbit.

-   -   16. A method according to paragraph 15 wherein the second Δv        increment has a value        Δv ₂ =v _(cs) −v _(ba) −Δv ₁.    -   17. A method for a vehicle launch from a first celestial body        and to an orbit about a second celestial body, said method        comprising        -   a. launching the vehicle using an impulsive force to provide            a first vehicle path;        -   b. applying a second force to the vehicle along a hyperbolic            path about the second celestial body and establishing a            circular orbit about the second celestial body.    -   18. A method according to any paragraph of paragraphs 10-17        wherein the method additionally comprises launching the vehicle        in an easterly direction.    -   19. A method according to paragraph 18 wherein the direction is        due east.    -   20. A method according to any of paragraphs 17-19 wherein the        first celestial body and the second celestial body are each        individually in orbit about a third celestial body.    -   21. A method according to paragraph 20, wherein the third        celestial body is the Sun.    -   22. A method according to any of paragraphs 17-19, wherein the        second celestial body is the Sun, the vehicle occupies a        LaGrange point of the first celestial body and the second        celestial body, and the LaGrange point is in said circular orbit        about the second celestial body.    -   23. A method for a vehicle launch from a first celestial body        and to a second celestial body, said method comprising        -   a. launching the vehicle at a latitude of about 23.4 deg            north or about 23.4 deg south in a direction due east using            a first force to establish a first vehicle path that is            hyperbolic about the first celestial body, said first force            comprising an impulsive force;        -   b. applying a second force to the vehicle to establish a            second vehicle path that intersects a path of the second            celestial body; and        -   c. applying a third force to the vehicle to establish the            vehicle in an orbit about the second celestial body.    -   24. A method according to paragraph 23 wherein the second force        places the second vehicle path in a plane of the second        celestial body's orbit.    -   25. A method according to any paragraph above wherein the        impulsive force is provided by a light gas gun.    -   26. A method according to any of paragraphs 1-24 wherein the        impulsive force is provided by an electromagnetic launcher,    -   27. A method according to any paragraph above wherein the        impulsive force is provided by a land-based or a sea-based        impulsive launcher.    -   28. A method of launching a vehicle to orbit about Earth,        wherein the method comprises launching the vehicle in a        direction that is easterly using an impulsive force and into a        path of a ballistic elliptical trajectory,        -   a. wherein the path of the ballistic elliptical trajectory            has a ballistic trajectory apogee and a ballistic trajectory            perigee and        -   b. wherein said direction of launch is in a plane            corresponding to the ballistic elliptical trajectory of the            vehicle.    -   29. A method according to paragraph 28 wherein the direction is        due east,    -   30. A method according to paragraph 28 or paragraph 29 wherein        the ballistic trajectory of the vehicle and an orbit of a space        object are in the same plane.    -   31. A method according to any one of paragraphs 28-30 wherein        -   a. the ballistic trajectory apogee is closely matched to (1)            an orbital altitude of a space object or (2) a place of            rendezvous with the space object,        -   b. wherein the vehicle has a fly-out time from a launch            site, said fly-out time being measured from a launch time to            a time that the vehicle first achieves the ballistic            trajectory apogee, and        -   c. wherein the vehicle launch occurs about one-third of said            fly-out time prior to the space object passing overhead of a            position of the launch site at the launch time.    -   32. A method according to any one of paragraphs 28-30 wherein        the method further comprises applying a first force to the        vehicle at the ballistic trajectory apogee, said force being        less than a force needed to establish a circular orbit for the        vehicle so that the vehicle enters a second and descending        elliptical trajectory which has a perigee at an altitude above        the Earth that is lower than an altitude of the ballistic        trajectory apogee.    -   33. A method according to paragraph 32 wherein the method        further comprises applying a second force to the vehicle at the        second elliptical trajectory perigee to establish a circular        orbit having an altitude lower than the altitude of the        ballistic trajectory apogee.    -   34. A method according to any one of paragraphs 28-30 wherein        the method further comprises applying a first force to the        vehicle at the ballistic trajectory apogee so that the vehicle        enters a second and ascending elliptical trajectory having an        apogee at an altitude above the Earth that is higher than an        altitude of the ballistic trajectory apogee.    -   35. A method according to paragraph 34 wherein the method        further comprises applying a second force to the vehicle at the        second elliptical trajectory apogee to establish a circular        orbit at an altitude above the Earth that is greater than the        altitude of the ballistic trajectory apogee.    -   36. A method according to any one of paragraphs 28-30 wherein        the method further comprises applying a first force to the        vehicle at the ballistic trajectory apogee to establish a        circular orbit for the vehicle.    -   37. A method for a vehicle launch from a first celestial body to        an orbit about a LaGrange point, said method comprising        -   a. launching the vehicle using an impulsive force to provide            a first vehicle path;        -   b. applying a second force to the vehicle along a hyperbolic            path about the LaGrange point and establishing an orbit            about the LaGrange point.    -   38. A method according to paragraph 37 wherein the first        celestial body and the LaGrange point are each individually in        orbit about a second celestial body.    -   39. A method, according to paragraph 38 wherein the first        celestial body is a planet and the second celestial body is the        Sun.    -   40. A method according to paragraph 37 wherein the LaGrange        Point and a second celestial body are each individually in orbit        about the first celestial body.    -   41. A method according to paragraph 40 wherein the first        celestial body is a planet and the second celestial body is a        moon of the first celestial body.

What is claimed is:
 1. A method of closing a timing difference between avehicle launched using an impulsive force and a satellite of rendezvousor a desired vehicle location, comprising applying a series of forces tothe vehicle to provide a change in vehicle velocity Δv that is dividedinto a first Δv increment and a second Δv increment, wherein a. a firstforce of the series provides the first Δv increment and temporarilyplaces the vehicle into a first orbit having a first orbital period thatdiffers from an orbital period of the satellite or the desired vehiclelocation so as to reduce a time difference between the vehicle and thesatellite or the desired vehicle location in an integer number oforbits, and b. a second force of the series provides the second Δvincrement and is sufficient to establish the vehicle in a circular orbitwith the satellite or the desired vehicle location, and wherein c. thevehicle follows a path of an elliptical trajectory having a ballisticapogee and a ballistic perigee; d. the first force is applied to thevehicle at the ballistic apogee to establish a second elliptical orbithaving a second apogee and a second perigee, the second elliptical orbitbeing a descending elliptical orbit, wherein the second apogee has anelevation equal to an elevation of the ballistic apogee, and e. thesecond force is applied when the vehicle is at the second apogee.
 2. Themethod of claim 1, wherein the second force additionally matches thevehicle velocity to a velocity of the satellite or the desired vehiclelocation.
 3. The method of claim 1, wherein the first Δv increment isselected to satisfy the equation ΔT=N(T_(cs)−T) where ΔT is a positivevalue that represents a timing difference between the satellite or thedesired vehicle location and the vehicle, T_(cs) is a period of thesatellite's orbit, T is a period of the elliptical orbit of the vehicle,and N is an integer number of orbits required to correct a distancebetween the vehicle and the satellite or the desired vehicle location.4. The method of claim 3, wherein the first Δv increment has a value${\Delta\; v_{1}} = {{- v_{ba}} + {v_{cs}\lbrack {2 - ( {{NT}_{cs}/( {{NT}_{cs} - {\Delta\; T}} )} )^{\frac{2}{3}}} \rbrack}^{\frac{1}{2}}}$where v_(ba) is a vehicle speed at ballistic apogee and v_(cs) is aspeed of the satellite in circular orbit.
 5. The method of claim 4,wherein the second Δv increment has a value Δv₂=v_(cs)−v_(ba)−Δv₁. 6.The method of claim 1, wherein the method additionally compriseslaunching the vehicle due east or in an easterly direction.
 7. Themethod of claim 1, wherein the impulsive force is, provided by a lightgas gun, an electromagnetic launcher, or a land-based or sea-basedimpulsive launcher.
 8. The method of claim 1, wherein the first orbitand the second orbit are coplanar.
 9. The method of claim 1, wherein thefirst force results from a first short motor burn and the second forceresults from a second short motor burn.
 10. The method of claim 1,wherein the vehicle has a launch trajectory, and the first force isapplied to the vehicle during the launch trajectory to establish thefirst orbit.
 11. A method of closing a timing difference between avehicle launched using an impulsive force and a satellite of rendezvousor a desired vehicle location, comprising applying a series of forces tothe vehicle to provide a change in vehicle velocity Δv that is dividedinto a first Δv increment and a second Δv increment, wherein a. a firstforce of the series provides the first Δv increment and temporarilyplaces the vehicle into a first orbit having a first orbital period thatdiffers from an orbital period of the satellite or the desire vehiclelocation so as to reduce a time difference between the vehicle and thesatellite or the desired vehicle location in an integer number oforbits, and b. a second force of the series provides the second Δvincrement and is sufficient to establish the vehicle in a circular orbitwith the satellite or the desired vehicle location wherein c. thevehicle follows a path of an elliptical trajectory having a ballisticapogee and a ballistic perigee; d. the first force is applied to thevehicle at the ballistic apogee to establish a second elliptical orbithaving a second apogee and a second perigee, the second elliptical orbitbeing an ascending elliptical orbit, wherein the second perigee has anelevation equal to an elevation of the ballistic apogee, and e. thesecond force is applied when the vehicle is at the second perigee. 12.The method of claim 11, wherein the second force additionally matchesthe vehicle velocity to a velocity of the satellite or the desiredvehicle location.
 13. The method of claim 11, wherein the first Δvincrement is selected to satisfy the equation ΔT=N(T_(cs)−T) where ΔT isa negative value that represents a timing difference between thesatellite or the desired vehicle location and the vehicle, T_(cs) is aperiod of the satellite's orbit, T is a period of the elliptical orbitof the vehicle, and N is an integer number of orbits required to correcta distance between the vehicle and the satellite or the desired vehiclelocation.
 14. The method of claim 13, wherein the first Δv increment hasa value${\Delta\; v_{1}} = {{- v_{ba}} + {v_{cs}\lbrack {2 - ( {{NT}_{cs}/( {{NT}_{cs} - {\Delta\; T}} )} )^{\frac{2}{3}}} \rbrack}^{\frac{1}{2}}}$where v_(ba) is vehicle speed at ballistic apogee and v_(cs) is a speedof the satellite in circular orbit.
 15. The method of claim 14, whereinthe second Δv increment has a value Δv₂=−(v_(cs)−v_(ba)−Δv₁).
 16. Themethod of claim 11, wherein the vehicle has a launch trajectory, and thefirst force is applied to the vehicle during the launch trajectory toestablish the first orbit.
 17. The method of claim 16, wherein thelaunch trajectory re-intersects the Earth.
 18. The method of claim 11,wherein the first force results from a first short motor burn and thesecond force results from a second short motor burn.
 19. A method ofclosing a timing difference between a vehicle launched using animpulsive force and a satellite of rendezvous or a desired vehiclelocation, comprising applying a series of forces to the vehicle toprovide a change in vehicle velocity Δv that is divided into a first Δvincrement and a second Δv increment, wherein a. a first force of theseries is applied to the vehicle during a launch trajectory whichreintersects Earth and which provides the first Δv increment thattemporarily places the vehicle into a first orbit having a first orbitalperiod which differs from an orbital period of the satellite or thedesired vehicle location so as to reduce a time difference between thevehicle and the satellite or the desired vehicle location in an integernumber of orbits, and b. a second force of the series provides thesecond Δv increment and is sufficient to establish the vehicle in acircular orbit with the satellite or the desired vehicle location, andwherein c. the launch trajectory is an elliptical trajectory having aballistic apogee and a ballistic perigee; d. the first force is appliedto the vehicle at the ballistic apogee to establish a second ellipticalorbit having a second apogee and a second perigee, the second ellipticalorbit being a descending elliptical orbit, wherein the second apogee hasan elevation equal to an elevation of the ballistic apogee, and e. thesecond force is applied when the vehicle is at the second apogee. 20.The method of claim 19, wherein the second force additionally matchesthe vehicle velocity to a velocity of the satellite or the desiredvehicle location.
 21. The method of claim 19, wherein the first forceresults from a first short motor burn and the second force results froma second short motor burn.